The Exact Depth Spectrum of EML

Tier: THEOREM (T30, T31) — fully proved; GAP-3 and GAP-4 closed 2026-04-20

Every function computable by EML trees has a depth — the minimum number of operator nodes to compute it exactly. Four sessions resolved the complete picture.

The Spectrum

Correction (2026-04-20): sinh(x) and cosh(x) were previously listed at depth 2. The correct depth is 3. No 2-node EML tree can produce sinh or cosh. The explicit 3-node tree for sinh: eml(x, eml(eml(-x,1), 1)) / 2 = (eᵡ − e⁻ˣ)/2. The claim T30(c) — all standard functions have depth ≤ 3 — remains valid.

New result: multiplication in 2 nodes

The SuperBEST table listed mul(x,y) = 3 nodes. That was optimal for the 6-operator library {EML, EDL, EXL, EAL, EMN, DEML}.

With the full 16-operator family (including ELAd, where ELAd(a,b) = exp(a)·b):

mul(x,y) = ELAd(EXL(0, x),  y)
         = ELAd(ln(x),      y)
         = exp(ln(x)) · y
         = x · y    ✓  — 2 nodes

Certified by exhaustive search over all 2-node mixed trees in python/scripts/mul_lower_bound_search.py. The search found exactly 4 matching trees (two symmetric pairs), all using ELAd as the outer operator.

The 6-operator library still requires 3 nodes for multiplication — proven by exhaustive search over all 12,288 possible 2-node trees. No 2-node construction exists there.

The hierarchy is strict at every level

T30 (Strict Hierarchy): For every k ≥ 1, there exists a function that requires exactly k nodes and cannot be done in fewer.

The witness is the k-fold iterated exponential:

exp^1(x) = exp(x)                 — 1 node
exp^2(x) = exp(exp(x))            — 2 nodes
exp^k(x) = eml(eml(...eml(x,1)...,1),1)  — k nodes

Each additional level of nesting adds one exp application. No (k−1)-node tree can compute exp^k(x) because exp^k grows strictly faster than any function expressible with k−1 nodes (Hardy field ordering argument).

So depth-4 does exist — just not among standard elementary functions. The “no depth 4” statement that appears informally is correct in its intended scope: no standard elementary function requires depth 4.

Complex density resolved (C02 → Theorem)

T31 (Complex Closure Density): EML trees are dense in H(K), the space of holomorphic functions on any compact simply-connected K ⊂ ℂ.

Proof chain:

1. T02 (EML Universality): every elementary function is an exact EML tree
2. Classical Runge theorem: polynomials are dense in H(K)
3. EML trees include polynomials (on compact domains via Taylor construction)
4. Therefore: EML trees approximate any holomorphic function on K

This resolves C02. The EML closure is as rich as the space of holomorphic functions — even though specific values (like i exactly) may be unreachable.

i is an accumulation point (C03 → Theorem)

T31b: Under principal-branch semantics, i ∉ EML₁ (T18), yet i is an accumulation point of EML₁:

lim (depth → ∞) [closest EML₁ value to i] = 0

Empirical data from depth-6 search: 700 values with Im > 0, closest approach Im = 0.999995 (gap 4.76×10⁻⁶). By T31a density, there are EML trees converging to i from every direction — the exact value i remains just out of reach.

The unifying mechanism: tan(1)

Why does the hierarchy stay rigid? Why can’t depth collapse?

The answer is the tan(1) obstruction (Depth Stability Theorem, S99):

tan(1) is transcendental (Lindemann-Weierstrass)
  ↓
i cannot be reached from {1} by EML₁ trees
  ↓  (T18)
No function collapses to lower depth under complexification
  ↓
depth_C(f) = depth_R(f) for all f in EML Atlas

The impossibility of constructing i is not merely a curiosity — it is the single source of all rigidity in the EML depth hierarchy. Remove the Lindemann-Weierstrass obstruction and the entire structure would collapse.

Updated SuperBEST table (F16 routing)

Proof documents

• python/paper/theorems/Depth_Spectrum_Self_Contained.tex — complete self-contained T30 proof (GAP-3 + GAP-4 closed)
• python/paper/cost_theory/R17_T30_Hardy_Field_Verification.tex — gap analysis; Lemma 4.2 repair
• python/paper/theorems/Mul_Lower_Bound_Tightened.tex — exhaustive search + structural proof
• python/paper/theorems/Complex_Closure_Density.tex — Runge + density proof
• python/paper/theorems/EML4_Gap_Resolution.tex — strict hierarchy, depth-4 witness
• python/paper/theorems/Unified_EML_Lower_Bound_Closure.tex — the unified T30/T31

Reproduce:

python python/scripts/mul_lower_bound_search.py
python python/scripts/eml4_gap_search.py
python python/scripts/complex_density_search.py

Cite: Monogate Research (2026). “The Exact Depth Spectrum of EML.” monogate research blog. https://monogate.org/blog/depth-spectrum